Math of frets

Bassguy87564 · #1 02-08-2008, 12:51 AM

Okay I will first start off by saying I'm a big math and physics guy, I love them almost as much as music. I was was just thinking one day about the math of find where the frets go on the neck. The first thing that came to mind was just a ratio with frequency and length of the neck. So I tried just that 55hz/110hz=32in/x, I did this first cause I know that it should be 1/2 the length. I looked at it then I realize that it would double the length so it wasn't right. Then I looked over at my bass on its stand and the first thing that came to my mind after looking the the way the frets are placed was a graph of an inverse function

(told you im a geek when it comes to this stuff also that is not my bass I have Dean Razor). So I inversed the length of the equation 55hz/110hz=1/32in/x or just 55hz/110hz=x/32in(x=16in). So far I think its coming out right is the first fret on a 32in scaled bass about 30.20422172645in away from the saddle(55hz/58.27hz=x/32in)? and do people use a similar equation? or do people use the proportions Pythagoras found for the interval of frequencies?

Siegy · #2 02-08-2008, 12:56 AM

...my brain is melting.

WarriorJoe7 · #3 02-08-2008, 01:02 AM

DistanceFromBridge = ScaleLength/(2^(FretNumber/NotesInOctave))

or for a standard 34 inch bass...

DistanceFromBridge = 34/(2^(FretNumber/12))
Fret number zero is the nut itself.

B.C. · #4 02-08-2008, 01:07 AM

Awsome man, I am a physics lover too!

lemur821 · #5 02-08-2008, 01:18 AM

I think most luthiers use a template (someone will be along to correct me, I'm sure).

People do not use a Pythagorean tuning. We use a scale with 12 equal steps to the octave. As for the math, an octave = twice the pitch, two octaves = four times, and so on. In other words where f is your starting frequency,

one octave = f*2^1
two octaves = f*2^2.

That's the same as
one octave = f^(12/12)
two octaves = f^(24/12),
right?

You just have to change the numerator of that fraction to the number of semitones above the root you want to find the note's sounding pitch.

Of course, to double the frequency of a vibrating string, you halve its length rather than doubling it. Where p is the distance from the bridge to a given interval and l is the length of the neck,

one octave: p = l/2 = l/2^(12/12)
two octaves: p = l/4 = l/2^(24/12)

Usually you want to measure from the nut rather than the bridge, so you subtract the distance above from the total neck length, l. p = l - l/2^(x/12), where x is the fret number, counting from the nut. Of course, this generalizes to any number of equal divisions of the octave. You want 30 out of 72 steps on a 34-inch neck? p = 34 - 34/2^(30/72) = 8.528 inches. A fourth: exactly what we wanted.

If you wanted to calculate the fret locations for a non-equal tuning, you would replace 2^(x/y) with the frequency ratio for each pitch in the scale. E.g., the note a fourth (4:3) up from the open string on a 34-inch neck would be 34 - 34/(4/3) = 8.5 inches.

WarriorJoe7 · #6 02-08-2008, 01:22 AM

Just made this Fret Calculator Excel Spreadsheet...

You just change the scale length and the number of notes in an octave (for like Indian Music or Middle Eastern Music etc) to whatever you want and the spreadsheet adjust all the numbers for the frets. This will handle ONLY equal temperament scales (Our American and most modern Western music scales are equal temperament.)

Al Heeley · #7 02-08-2008, 03:45 AM

I like this discussion. I want to throw in another complication. The tuning on a guitar is 'approximate', best compromise, hence the evolution of the Earvana compensated nut and the Buzz Feiten tuning system which is supposed to compensate for string length/tone. I don't understand what the problem is or how these compensated nuts help - it shouldn't be related to string diameter, it must be length/tension. All I know is that on a 6-string the B and the G played open seldom sound bang-on, or if they do, the moment a chord is fretted, they sound a fraction off again.

pilotjones · #8 02-08-2008, 11:29 AM

I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS. I will not get started about BFTS.

Bassguy, if you're a math/physics guy too, you can get all the relevant equations from the formulas in this spreadsheet.
String Calculation Spreadsheet - please comment

lemur821 · #9 02-08-2008, 12:28 PM

Quote:

Originally Posted by Al Heeley

I don't understand what the problem is or how these compensated nuts help - it shouldn't be related to string diameter, it must be length/tension. All I know is that on a 6-string the B and the G played open seldom sound bang-on, or if they do, the moment a chord is fretted, they sound a fraction off again.

As I understand it, the BFTS compensates for the [what should be well known] deficiencies of a scale composed of 12 equal steps. Here's a video where he talks about it:
http://gedgreen.co.uk/fileadmin/vid3.html

Some of his wordings are odd. He seems to consider the 12-tone fourth, fifth, third, &c. to be what you might call the "mathematically proper" tuning, and he talks about the ear being more tolerant of deviations from the perfect intervals, such as the fifth and fourth, than it is of deviations from the third and sixth.

Actually, I don't think that's true at all. In the 12-tone scale, the fourths and fifths are extremely close to their purely tuned counterparts. The thirds and sixths, on the other hand, are around 15 cents (15/100 of a semitone) from where they ought to be! A world that tunes religiously to the 12-tone scale suggests that people are very tolerant of mistuned thirds and sixths, and prefer to have their fifths and fourths in tune.

However, some people are (rightfully) bothered by those mistuned intervals. The BFTS worsens the fifths and fourths slightly in order to improve the thirds and sixths slightly. I believe the effect is not quite constant across the neck, so that the exact degree of correction varies depending on where you play. I'm just going by the video with regard to the BFTS; I have no hands-on experience with it.

pilotjones · #10 02-08-2008, 12:49 PM

Looking at the patents for BFTS, my interpretation is that it has nothing to do with resolving 12TET temperament deficiencies. Anything that attempts to do so (I believe) necessarily creates an instrument that sounds better in one key and worse in others. I believe BFTS is a combination of compensated nut, stretch tuning, and slightly off intonation.

eleonn · #11 02-08-2008, 02:21 PM

Ehhhhhhh... what does BFTS means?

asad137 · #12 02-08-2008, 02:44 PM

Quote:

Originally Posted by eleonn

Ehhhhhhh... what does BFTS means?

Buzz Feiten Tuning System

grace & groove · #13 02-08-2008, 02:48 PM

Buzz Feiten Tuning System.

And fretless ftw.

lemur821 · #14 02-08-2008, 04:11 PM

Quote:

Originally Posted by pilotjones

Looking at the patents for BFTS, my interpretation is that it has nothing to do with resolving 12TET temperament deficiencies. Anything that attempts to do so (I believe) necessarily creates an instrument that sounds better in one key and worse in others. I believe BFTS is a combination of compensated nut, stretch tuning, and slightly off intonation.

I made my assumption based on the way Feiten described it in the video. With all his talk about altering the perfect intervals in order to get flatter thirds, I don't see what else it could be. He's very explicit. I think its detrimental effects on certain keys are mitigated by the fact that it mostly (just?) acts on the G and B strings.

WarriorJoe7 · #15 02-08-2008, 05:46 PM

I could very well be wrong, but I heard that the BFTS does an adjustment because when you fret a note you lengthen the string. Because the lengthening of the string is slightly different for different frets (see my pic below,) the intonation is not perfect. If you wanted to have a bass with perfect intonation you would have to design it so that instead of pressing the string down you bring the fret up when you wanted to play a note. The BFTS is supposed to adjust for this.

However, the lower you have your action at the first fret (based on the depth of nut slots and if the nut is correctly cut) the less problem you have with this. The other thing that puzzles me is that depending on the depth of the nut slots (which affects the distance between the first fret and the string when it's not fretted... the gap) it seems to me that you would need a slightly different BFTS for each bass based on the size that first fret gap. That is unless once you install it you are supposed to set it up to a standardized gap (but even with this you would still need to do this for every fret which means that you have to have the correct bridge saddles height and this one is the kicker... the correct curvature in the neck. This is akin to trying to make one size and shape pair of pants fit everyone in the world since truss rods don't adjust every neck witht the same curvature.) So if this is what the BFTS is for, it would get no closer to perfect tuning for the standard bass than not having it. That is unless someone designed a seperate BFTS for every bass.

OK so here is my pic that shows why the string is lengthened different amounts based on which fret you press the string to. I exagerated the height of the nut in order to show the problem more easily.

Son of Magni · #16 02-08-2008, 06:03 PM

There's a free downloadable app called wfret (google...) that has a couple different ways of calculating it...

It gives the distance from nut, distance between frets, and prints a template.

Bassguy87564 · #17 02-08-2008, 06:12 PM

Thanks lemur821 for that equation and Thank you WarriorJoe7 for the excel. My way does work but its to find the distance from the bridge. Also I do understand the excel sheet pilotjones. I also wanted to ask how did you come about this equation, Its interesting also looks helpful for every string instrument.

Son of Magni · #18 02-08-2008, 06:19 PM

Quote:

Originally Posted by pilotjones

Looking at the patents for BFTS, my interpretation is that it has nothing to do with resolving 12TET temperament deficiencies. Anything that attempts to do so (I believe) necessarily creates an instrument that sounds better in one key and worse in others. I believe BFTS is a combination of compensated nut, stretch tuning, and slightly off intonation.

Don't get us started on stretch tuning

T2W · #19 02-08-2008, 07:54 PM

Im sure I would have to read this thread numerous times to understand exactly what's going on, but this has got to be the most interesting thread on TB ive read so far. subscribed. Is it possible, however, that for a bass, its not as important (please dont throw me any stones) since bassists dont do (or most of the time) many chords? I mean surely bassists jump from string to string, note to note, but chances of hearing a big difference are'nt as great as with a g****r (stones again, please) since two strings are rarely played at the same time? I need to take out my school books and stay up till 4am....

WarriorJoe7 · #20 02-09-2008, 01:15 PM

Also wanted to clarify that there are 2 equations you can use depending on whther you want the distance between the fret and the nut or between the fret and the bridge.

I already did incorporate this into the spreadsheet I made but here are the formulas I used to make the spreadsheet...

DistanceFromBridge = ScaleLength/(2^(FretNumber/NotesInOctave))
DistanceFromNut = ScaleLength-ScaleLength/(2^(FretNumber/NotesInOctave))

or for a standard 34 inch bass...

DistanceFromBridge = 34/(2^(FretNumber/12))
DistanceFromNut = 34-34/(2^(FretNumber/12))
Fret number zero is the nut itself.